Yeah e-ink uses no power, only when it refreshes. If you refresh say from 7 AM till 11 PM but not in night then WLAN has to fire up (cheap on ESP32 etc) and you could easily use a poor man's powerbank such as 18650 in USB enclosure.

Tho I think PoE is neat for a device like this. White cable (assuming wall is white) and you're done.

Though I wouldn't rely on a proprietary solution. Nor on Google Calendar.

There are plenty of such options available. For example [1] but there's many more. Here another [2]. Can't find the one I got but yeah it broke and remember you need to update the software at times. I put it on a VLAN and guest WLAN to be sure.

[1] https://hackaday.io/project/189632-e-ink-weather-station-pum...

[2] https://www.hackster.io/lmarzen/esp32-e-paper-weather-displa...